Mathematical proficiency
Tying it all together : Weaving in
the strands of Mathematical
Proficiency
1. Conceptual understanding –
2. Procedural fluency –
3. Strategic competence –
4. Adaptive reasoning5. Productive disposition –
Jennifer Suh
George Mason University
[email protected]
Presentation can be found on
http://mason.gmu.edu/~jsuh4/teaching/strands.htm
Strands of Mathematical Proficiency
Kettle Math- example of what it is not.
Mathematical Proficiency
• conceptual understanding - comprehension of
mathematical concepts, operations, and
relations
• procedural fluency - skill in carrying out
procedures flexibly, accurately, efficiently, and
appropriately
• strategic competence - ability to formulate,
represent, and solve mathematical problems
• adaptive reasoning- capacity for logical thought,
reflection, explanation, and justification
• productive disposition - habitual inclination to
see mathematics as sensible, useful, and
worthwhile, coupled with a belief in diligence
and one's own efficacy.
The Forgotten Standards…
NCTM Process Standard
• Problem Solving
• Reasoning and Proof
• Communication
• Connections
• Representations
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CONCEPTUAL UNDERSTANDING
Conceptual Understanding
• MODELING MATHEMATICS
Concept maps
Taboo
Pictionary
wiki-board
http://mathbridges.pbwiki.com/
Fraction
1)
Region Model( Pattern Blocks) most
concrete for students to understand. The
region is the whole and the parts are congruent
(same size and shape) Types of regions include
circles, rectangle, square, triangle
Fraction Folding (FD 7,8,9,10)
Pattern Block (FD 35, 36, & FD 49-56)
Length model
2)
Length model( Tape measures) Any unit of
length can be partitioned into fractional parts
(Number line, ruler, long strip of paper,
cuisinaire blocks)
• Cusinaire activity ( Cusinaire riddles)
• Number Line Fraction (Fraction Bar game)
Fraction Circles(Equivalent Fraction –Aims)
2
Set model
Area model
3)
4)
set model ( 2 color counters) uses a set of objects as
whole; This sometimes causes difficulty because
students don't see the set of objects as one complete
whole. Children need many experience partitioning sets(
background for division) then seeing if the sets can be
partitioned into equal parts
Area model( geoboards) more sophisticated
version of the region model. The parts must
have equal area but do not have to be in the
same shape, Students need an understanding
of area.
• Color Counter activity (page FD3,4,5)
• Four bean salad
Strategic Competence
• Ability to formulate, represent, and solve
mathematical problems
• Experience with problem solving strategies
• Flexibility of approaches
Geoboard activity
Math Happening
• OBJECTIVE: Share a real-life event (math
happening) and pose a question that can be
answered using the information given in the
story. Illustrate the number sentence by
drawing a picture.
Problem Posing
Math Curse…at Westlawn?
Strategy cards
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Adaptive reasoning
• capacity for logical thought, reflection, explanation, and
justification
Better Buy… adaptive reasoning
I went shopping to Costco and to Giant
with my bargain shopper husband
and compared prices on our snacks.
What is a better buy?
Choose a better buy.
1) 4 oz can of peanuts for 45 cents
10 oz can of peanuts for 90 cents
2) 14oz sauce for 79cents
18oz sauce for 81 cents
3) 3oz bag of chips for 30cents or
4 oz bag for 44cents
Procedural Fluency
How does the lattice
multiplication work?
http://nlvm.usu.edu/en/nav/frames_asid_192_g_2_t_1.html
procedural fluency - skill in
carrying out procedures flexibly,
accurately, efficiently, and
appropriately
Challenge 24
Count down
Stand up, hand up and share
Procedural fluency
Mental Computation
Rounding and Number Line
Materials: Clothesline and clothespin
Hanger
• Observing and using relationships
• Using patterns
• Using thinking strategies
• Recognizing different forms
• Connect number words and numerals to the quantities they
represent, using various physical models and representations
Estimation
• Judging unreasonableness
• Using numbers flexibly
• Using referents and benchmarks
• Relative position and magnitude of whole/rational numbers
4
Fraction Line up
Judging Reasonableness…
• Close to 0, ½ and 1
•
Fraction circles or fraction bar
•
Fraction mat with 0-1/2 – 1
• Number Sense with Fraction & Decimals
• Reflection questions:
• What’s alike about all the fractions close to 1?
• What’s alike about all fractions close to ½?
• What’s alike about all fractions close to 0?
Problem of the week
http://www.figurethis.org/
http://eduplace.com
Literature Connections
Present math activities in
contexts.
Assessing math proficiency
• Publication Data:
Adding It Up: Helping
Children Learn
Mathematics,
Mathematics Learning
Study Committee,
Jeremy Kilpatrick, Jane
Swafford, and Bradford
Findell, editors. National
Academy Press, 2001.
Hardcover, 480 pp,
$29.95. ISBN 0-30906995-5.
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